You're right to avoid writing essentially the same answer, but if you think you can write a better answer than one of the existing answers, you should definitely go ahead.
The primary purpose of Stack Overflow is to create a resource of great answers. It's not designed to be a support forum, so it doesn't matter whether there's a solution, it matters whether there's a great solution. Perhaps you can explain something much clearer than the existing three-line quick answer.
The best way of getting rep is to write really good answers.
You can sort the Unanswered questions list by using the newest tab. Once a question has fallen off the front page or two, it's much less visible to the instant rep seekers.
You can also cut out a lot of the noise by searching a tag you're particularly expert in - you'll see older questions there that are off the radar of the front page. I tend to browse only one or two tags at a time and hardly ever visit the homepage. It's clearer then who's not getting an answer, and sometimes it's because no-one who knows can be bothered explaining or looking into it in detail. That's a good opportunity for you. (You can add favourite tags on the right hand side and they'll be highlighted in the general list so they stand out for you.)